Almost all $C_k$-free oriented graphs have $Θ(n)$ backwards edges

Abstract

We prove a conjecture of Kühn, Osthus, Townsend and Zhao \cite{kuhn2017structure} stating that almost every $C_k$-free oriented graph on $n$ vertices has $Θ(n)$ backwards edges in a transitive-optimal ordering. The same holds for $C_k$-free digraphs when $k$ is even. Our proof combines the hypergraph container method with a stability analysis and an inductive counting argument. As a byproduct, we also determine the typical structure of oriented graphs and digraphs that avoid the blow-up $C_{k}^t$, extending the main result of \cite{kuhn2017structure} to the blown-up setting.